Casimir effect associated with fractional laplacian and fractal dimensions

Casimir effect predicts that two parallel flat neutral plates are attracted to each other due to quantum fluctuations of the electromagnetic field. In this communication, we study Casimir effect based on two different approaches: the first one is correlated to fractional Laplacian by means of fractional derivatives operators whereas the second one is associated to fractal dimensions. It was observed that, in both approaches, the Casmir zeropoint energy of the fields is affected by the fractional order parameter and the numerical value of the fractal dimensions. The first model requires the use of the Hurwitz zeta function and no regularization technique is strongly required. Nevertheless, the second model necessitates the use of the regularization technique augmented by the Lifshitz fixed point and Schwinger proper-time representation mainly when a particular eigenvalue problem is used.